The matrix norm induced by a published vector p-norm

Let $m,n\in \mathbb{N}$, let $p\in \mathbb{Q}$ with $p\ge 1$, and let $A=(a_{ij})\in M_{m\times n}(\mathbb{R})$ be a real matrix (Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes), regarded as the linear map $x\mapsto Ax$ from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m$ with the published $p$-norms (The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$). The matrix $p$-norm induced by $\parallel \cdot {\parallel }_p$ is

$\parallel A{\parallel }_p:=\sup \{\parallel Ax{\parallel }_p:x\in {\mathbb{R}}^n,\parallel x{\parallel }_p\le 1\}\mathrm{.}$

When $n\ge 1$, the following three displayed quantities are the same number:

$\parallel A{\parallel }_p={\sup }_{\parallel x{\parallel }_p\le 1}\parallel Ax{\parallel }_p={\sup }_{\parallel x{\parallel }_p=1}\parallel Ax{\parallel }_p={\sup }_{x\mathrm{\ne }0}\frac{\parallel Ax{\parallel }_p}{\parallel x{\parallel }_p},$

the equalities following from positive homogeneity of $\parallel \cdot {\parallel }_p$ (Each $\parallel \cdot {\parallel }_p$ is a norm on ${\mathbb{R}}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_{\mathrm{\infty }}$ of the published metric-spaces page, axiom (N2)) and the equality $\parallel 0{\parallel }_p=0$.

The value is a finite nonnegative real. For $n\ge 1$ the norms $\parallel \cdot {\parallel }_p$ and $\parallel \cdot {\parallel }_2$ are equivalent by For $n\ge 1$ all norms on ${\mathbb{R}}^n$ are equivalent, so there is $C>0$ with $\parallel x{\parallel }_{\mathrm{\infty }}\le C\parallel x{\parallel }_p$ for every $x$. For every $i<m$, $\mathrm{\mid }(Ax{)}_i\mathrm{\mid }\le {\sum }_{j<n}\mathrm{\mid }{a}_{ij}\mathrm{\mid }\mathrm{\mid }{x}_j\mathrm{\mid }\le ({\sum }_{j<n}\mathrm{\mid }{a}_{ij}\mathrm{\mid })\parallel x{\parallel }_{\mathrm{\infty }}\le C({\sum }_{j<n}\mathrm{\mid }{a}_{ij}\mathrm{\mid })\parallel x{\parallel }_p$, so $\parallel Ax{\parallel }_p\le ({\sum }_{i<m}({\sum }_{j<n}\mathrm{\mid }{a}_{ij}\mathrm{\mid }{)}^p{)}^{1\mathrm{/}p}C\parallel x{\parallel }_p$; hence the supremum over the unit ball is at most the finite constant $({\sum }_{i<m}({\sum }_{j<n}\mathrm{\mid }{a}_{ij}\mathrm{\mid }{)}^p{)}^{1\mathrm{/}p}C$.

The induced infinity norm is a separate named case. When $m,n\ge 1$, define

$\parallel A{\parallel }_{\mathrm{\infty }}:=\sup \{\parallel Ax{\parallel }_{\mathrm{\infty }}:x\in {\mathbb{R}}^n,\parallel x{\parallel }_{\mathrm{\infty }}\le 1\}\mathrm{.}$

As in the rational-$p$ case, positive homogeneity gives

$\parallel A{\parallel }_{\mathrm{\infty }}={\sup }_{\parallel x{\parallel }_{\mathrm{\infty }}\le 1}\parallel Ax{\parallel }_{\mathrm{\infty }}={\sup }_{\parallel x{\parallel }_{\mathrm{\infty }}=1}\parallel Ax{\parallel }_{\mathrm{\infty }}={\sup }_{x\mathrm{\ne }0}\frac{\parallel Ax{\parallel }_{\mathrm{\infty }}}{\parallel x{\parallel }_{\mathrm{\infty }}}\mathrm{.}$

The supremum is finite because for every $i<m$ one has $\mathrm{\mid }(Ax{)}_i\mathrm{\mid }\le {\sum }_{j<n}\mathrm{\mid }{a}_{ij}\mathrm{\mid }\mathrm{\mid }{x}_j\mathrm{\mid }\le ({\sum }_{j<n}\mathrm{\mid }{a}_{ij}\mathrm{\mid })\parallel x{\parallel }_{\mathrm{\infty }}$, so

$\parallel Ax{\parallel }_{\mathrm{\infty }}\le {\max }_{i<m}{\sum }_{j<n}\mathrm{\mid }{a}_{ij}\mathrm{\mid }\parallel x{\parallel }_{\mathrm{\infty }}\mathrm{.}$

At $n=0$ the convention is explicit. The only element of ${\mathbb{R}}^0$ is the zero vector, the only matrix $A\in M_{m\times 0}(\mathbb{R})$ is the empty matrix, and the unit-ball definition gives $\parallel A{\parallel }_p=0$. The unit sphere and the set of nonzero vectors are empty, so the two ratio formulas above are not asserted in this case.

The induced quantity is a norm on the matrix space $M_{m\times n}(\mathbb{R})$: for each rational $p\ge 1$, and likewise for the separately named $\mathrm{\infty }$-case when $m,n\ge 1$, separation, absolute homogeneity and the triangle inequality follow from the same three axioms of the underlying vector norm (Each $\parallel \cdot {\parallel }_p$ is a norm on ${\mathbb{R}}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_{\mathrm{\infty }}$ of the published metric-spaces page) through the defining supremum. The compatibility with matrix-vector multiplication, the submultiplicativity and the value on identity matrices are Induced matrix norms are compatible with matrix-vector multiplication, submultiplicative, and satisfy ||I|| = 1; the explicit value at $p=1$ and the separate $\mathrm{\infty }$-formula are The induced 1-norm is the maximum column sum and the induced infinity-norm is the maximum row sum.

• At $p=2$ this is the published operator norm. The vector $2$-norm is the Euclidean norm (Each $\parallel \cdot {\parallel }_p$ is a norm on ${\mathbb{R}}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_{\mathrm{\infty }}$ of the published metric-spaces page), so the induced $2$-norm of a real matrix is the operator norm of the published spectral/SVD page; The operator norm is 0 on the zero domain and otherwise equals the largest singular value, attained at a right-singular vector identifies it with the largest singular value. No second operator norm is introduced here.

• Only rational $p\ge 1$ are used. The published The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$ exist for rational exponents only, and this definition inherits that restriction verbatim; no statement on this page ranges $p$ over a real interval.